Abstract
We formulate and analyze new methods for the solution of a partial integrodifferential equation with a positive-type memory term. These methods combine the finite element Galerkin (FEG) method for the spatial discretization with alternating direction implicit (ADI) methods based on the Crank---Nicolson (CN) method and the second order backward differentiation formula for the time stepping. The ADI FEG methods are proved to be of optimal accuracy in time and in the $$L^2$$L2 norm in space. Furthermore, the analysis is extended to include an ADI CN FEG method with a graded mesh in time for problems with a nonsmooth kernel. Numerical results confirm the predicted convergence rates and also exhibit optimal spatial accuracy in the $$L^{\infty }$$L? norm.
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